Conduction Law: Differential Form

Q.

Find the rate of heat flow through a cross section of the rod shown in figure $({\theta }_2>{\theta }_1)$. Thermal conductivity of the material of the rod is K.

Q.

A rod 1 m long and made of material of thermal conductivity 420 W/m/K has one of its ends in melting ice and the other end in boiling water. If its area of cross-section is 10 $c{m}^2$, the amount of ice that melts in 1 minute is

1. 0.125 g

2. 75 g

3. 7.5 g

4. 450 g

Q. A refrigerator door is $150\text{cm}$ high, $80\text{cm}$ wide and $6\text{cm}$ thick. If the coefficient of conductivity is $0.0005{\text{cal/cm-s-}}^{\circ }\text{C}$ and the inner and outer surfaces are at $0^{\circ }\text{C}$ and ${30}^{\circ }\text{C}$ respectively, find the heat loss per minute through the door. (in Calories)

1. $900$
2. $1200$
3. $3600$
4. $1800$

Q. A metallic spherical shell having inner and outer radii $a$ and $b$ respectively has thermal conductivity $K=\frac{K_0}{r^2}\{a\le r\le b\}$ The inner surface is maintained at temperature ${\theta }_1$ and outer surface is maintained at temperature ${\theta }_2$ . Calculate the rate at which heat flows radially through the material $[{\theta }_2>{\theta }_1]$.

1. $\frac{4\pi K_0({\theta }_1-{\theta }_2)}{b-a}$
2. $\frac{4\pi K_0({\theta }_2-{\theta }_1)}{b-a}$
3. $\frac{4\pi K_0(b-a)}{{\theta }_2-{\theta }_1}$
4. $\frac{8\pi K_0({\theta }_2-{\theta }_1)}{b-a}$

Q.

Two cylindrical rods of lengths $l_{1}and{l}_2$, radii $r_{1}and{r}_2$ have thermal conductivities $k_{1}and{k}_2$ respectively. The ends of the rods are maintained at the same temperature difference. If $l_1=2l_{2}and{r}_1=\frac{r_2}{2}$, the rates of heat flow in them will be the same if $\frac{k_1}{k_2}$ is

1. 1

2. 2

3. 4

4. 8

Q. An icebox made of 1.5 cm thick Styrofoam has dimensions $60cm\times 60cm\times 30cm$. It contains ice at $0^{\circ }C$ and is kept in a room at ${40}^{\circ }C$. Find the rate at which the ice is melting. Latent heat of fusion of ice = $3.36\times {10}^{5}J/kg$. and thermal conductivity of Styrofoam = $0.04W/m^{\circ }C$.

1. 0.316 g/s
2. 0.46 g/s
3. 1.46 g/s.
4. 0.66 g/s

Q. A cylindrical steel rod, $0.01m$ in diameter and $0.2m$ in length is first heated to ${750}^{\circ }C$and then immersed in water bath at ${100}^{\circ }C.$ The heat transfer coefficient is $250W/m^{2}K.$ The density, specific heat and thermal conductivity of steel are $\rho =7801kg/m^3,c=473J/kgK,$ and $k=43W/mK,$ respectively. The time required for the rod to reach ${300}^{\circ }C$ is seconds

Q.

A hole of radius $r_1$ is made centrally in a uniform circular disc of thickness d and radius $r_2$. The inner surface (a cyclinder of length d and radiud $r_1$) is maintained at a temperature ${\theta }_1$ and the outer surface (a cylinder of length d and radius $r_2$) is maintained at a temperature ${\theta }_2({\theta }_4>{\theta }_2)$. The thermal conductivity of the material of the disc is K. Calculate the heat flowing per unit time through the disc.

Q. A closed cubical box is made of perfectly insulating material and the only way for heat to enter or leave the box is through two solid cylindrical metal plugs, each of cross-sectional area $12c{m}^2$ and length 8 cm fixed in the opposite walls of the box. The outer surface of one plug is kept at a temperature of ${100}^{\circ }C$ while the outer surface of the other plug is maintained at a temperature of $4^{\circ }C$. The thermal conductivity of the material of the plug is $2.0W/m^{\circ }C$. A source of energy generating 13 W is enclosed inside the box. Find the equilibrium temperature of the inner surface of the box assuming that it is the same at all points on the inner surface.

1. 
2. 
3. 
4. 

Q. Two rods of same length but made of different material are taken for a study as shown in the figure. Area of cross-section of rod $1$ is $12{\text{cm}}^2$ and that of rod $2$ is $16{\text{cm}}^2$. Thermal conductivity of rod $1$ is $0.0008{\text{cal/cm-s-}}^{\circ }\text{C}$. If rate of loss of heat due to conduction is equal, find the thermal conductivity of rod $2$.

1. $0.0006{\text{cal/cm-s-}}^{\circ }\text{C}$
2. $0.0008{\text{cal/cm-s-}}^{\circ }\text{C}$
3. $0.0010{\text{cal/cm-s-}}^{\circ }\text{C}$
4. $0.0004{\text{cal/cm-s-}}^{\circ }\text{C}$

Q. The surroundings become warm when water in lakes start freezing in cold countries. Why?

Q. A cylindrical drum of height $1\text{m}$ made of a bad conductor is completely filled with water at $0^{\circ }\text{C}$ and is kept outside without a lid. The temperature of the surroundings is $-{10}^{\circ }\text{C}$. Calculate the time taken for the entire mass of the water to freeze. (Thermal conductivity of ice $=k$, latent heat of fusion $L_f=L)$
[Neglect expansion of water on freezing, $\rho$ = density of water]

1. $\frac{\rho L}{20k}$
2. $\frac{\rho L}{5k}$
3. $\frac{\rho L}{10k}$
4. $\frac{\rho Lk}{10}$

Q.

A hollow metallic sphere of radius 20 cm surrounds a concentric metallic sphere of radius 5 cm. The space between the two spheres is filled with a nonmetallic material. The inner and outer spheres are maintained at ${50}^{\circ }C$ and ${10}^{\circ }C$ respectively and it is found that 100 J of heat second. Find the thermal conductivity of the material between the spheres.

Q. A refrigerator door is $150\text{cm}$ high, $80\text{cm}$ wide and $6\text{cm}$ thick. If the coefficient of conductivity is $0.0005{\text{cal/cm-s-}}^{\circ }\text{C}$ and the inner and outer surfaces are at $0^{\circ }\text{C}$ and ${30}^{\circ }\text{C}$ respectively, find the heat loss per minute through the door. (in Calories)

1. $3600$
2. $1800$
3. $900$
4. $1200$

Q. An electric dipole is placed at an angle of ${30}^o$ with an electric field of intensity $2\times {10}^{5}N{C}^{-1}$. It experiences a torque of $4Nm$. Calculate the charge on the dipole if the dipole length is $2cm$ :

1. $8mC$
2. $4mC$
3. $8\mu C$
4. $2mC$

Q.

Three rods of identical cross-sectional area and made from the same metal form the sides of an isosceles triangle ABC, right angled at B. The points A and B are maintained at temperatures T and $\sqrt{2}T$ respectively. In the steady state, the temperature of point C is $T_c$. Assuming that only heat conduction takes place, the ratio $\frac{T_c}{T}$ is

1. 

2. 

3. 

4. 

Q. A cylindrical drum of height $1\text{m}$ made of a bad conductor is completely filled with water at $0^{\circ }\text{C}$ and is kept outside without a lid. The temperature of the surroundings is $-{10}^{\circ }\text{C}$. Calculate the time taken for the entire mass of the water to freeze. (Thermal conductivity of ice $=k$, latent heat of fusion $L_f=L)$
[Neglect expansion of water on freezing, $\rho$ = density of water]

1. $\frac{\rho L}{10k}$
2. $\frac{\rho Lk}{10}$
3. $\frac{\rho L}{20k}$
4. $\frac{\rho L}{5k}$

Q. A lake surface is exposed to an atmosphere where the temperature is less than $0$. If the thickness of the ice layer formed on the surface grown from 2 cm to 4 cm in 1 hour, the atmospheric temperature will be (Thermal conductivity office, $K=4\times {10}^{-3}cal{cm}^{-1}{s}^{-1}{}^{-1}$, density of ice $0.9g{cc}^{-1}$. Latent heat of fusion of ice $80cal{g}^{-1}$ Neglect the change of density during the state change. Assume that the water below the ice has $0$ temperature every where.)

1. $-20$
2. $0$
3. $-15$
4. $-30$

Q. A refrigerator door is $150\text{cm}$ high, $80\text{cm}$ wide and $6\text{cm}$ thick. If the coefficient of conductivity is $0.0005{\text{cal/cm-s-}}^{\circ }\text{C}$ and the inner and outer surfaces are at $0^{\circ }\text{C}$ and ${30}^{\circ }\text{C}$ respectively, find the heat loss per minute through the door. (in Calories)

1. $3600$
2. $1800$
3. $900$
4. $1200$

Q. A bimetallic strip is formed out of two identical strips one of copper and the other of brass. The co-efficients of linear expansion of the two metals are $a_C$ and $a_B$. On heating, the temperature of the strip goes $D_T$ and the strip bends to form an arc of radius of curvature R. Then R is

1. Proportional to
2. Inversely proportional to
3. Proportional to
4. Inversely proportional to

Q.

Water is densest at 4${}^{\circ }$C; this is the reason lakes do not completely freeze during extreme winters. Say, the atmosphere is at -${\theta }^{\circ }C(\theta >0)$, now the temperature of the lake's water starts to fall, and the denser water from the top sinks, getting the bottom layers up, and cooling them until the temperature reaches 4 ${}^{\circ }$C at the upper surface. Now, further reduction in temperature actually makes the water less dense! Hence the colder water stays on top, starts to freeze, until the top layer of ice reaches -${\theta }^{\circ }C$ and the bottom layer at 0 ${}^{\circ }$ C . And this layer of ice is the only area through which the water below loses heat (by conduction), slowly increasing the thickness of the ice (because ice is a bad conduction), and for a very long time the water at the bottom of the lake remains at 4 ${}^{\circ }$ C! Now, Let the thickness of the layer of ice be $y_1$, at an instant.How much time t, will it take to increase to thickness $y_2$?

$\rho \to$ density of the ice

K $\to$ thermal conductivity of ice.

L $\to$ Latent heat of fusion

$\theta \to$ temperature of surface (as mentioned earlier)

1. $\frac{\rho L}{K\theta }(y_2-y_1)$

2. $\frac{\rho L}{2K\theta }(y_2-y_1)$

3. $\frac{\rho L}{K\theta }(y_2^2-y_1^2)$

4. $\frac{\rho L}{2K\theta }(y_2^2-y_1^2)$

Q.

Two rods A and B of different materials are welded together as shown in the figure. If their thermal conductivities are $k_1$ and $k_2$, the thermal conductivity of the composite rod will be

1. $2(k_1+k_2)$

2. $\frac{3}{2}(k_1+k_2)$

3. $\frac{1}{2}(k_1+k_2)$

4. $k_1+k_2$

Q. Conductivity of water (in ${\Omega }^{-1}c{m}^{-1}$) is

1. 0.4
2. 0.04
3. 0.0004
4. 0.00004

Q. The channel having cross section given below carries a discharge of 10 cumecs. The flow is :

1. Subcritical
2. Supercritical
3. Critical
4. Can't determine

Q. A very thin metallic shell of radius r is heated to temperature t and then allowed to cool. the rate of cooling of shell is proportional to.

1. rT
2. $\frac{1}{r}$
3. $r^2$
4. $r^0$

Q. An open cubical tank was initially fully filled with water. When the tank was accelerated on a horizontal plane along one of its side it was found that one third of volume of water spilled out. The acceleration was

1. $g/3$
2. $2g/3$
3. $3g/2$
4. None

Q.

On a cold winter day, the atmospheric temperature is -$\theta$ (on Celsius scale) which is below 0${}^{\circ }$C. A cylindrical drum of height h made of a bad conductor is completely filled with water at 0 ${}^{\circ }$ C and is kept outside without any lid. Calculate the time taken for the whole mass of water to freeze. Thermal conductivity of ice is K, $\rho$ is the density and latent heat of fusion is L. Neglect expansion of water on freezing

1. 

2. 

3. 

4. 

Q.

List - I List - II

a. Temperature of water e.273 K

at the bottom of a lake

b.${\gamma }_r$of water is - ve f. 277 K

between

c. ${\gamma }_r$ of water is positive g. above 277 K

d. Temperature of water h. 273K and

just below ice layer in 277 K

a Lake

1. a- g, b-h, c-e, d-f
2. a-h, b-e, c-f, d-g
3. a-e, b-f, c-g, d-h
4. a-f, b-h, c-g, d-e

Q. A brass rod of length 2 m and cross-sectional area 2.0 $c{m}^2$ is attached end to end to a steel rod of length L and cross-sectional area 1.0 $c{m}^2.$ The compound rod is subjected to equal and opposite pulls of magnitude $5\times {10}^{4}N$ at its ends. If the elongations of the two rods are equal the length of the steel rod (L) is
($Y_{Brass}=1.0\times {10}^{11}N/m^{2}and{Y}_{Steel}=2.0\times {10}^{11}N/m^2$)

1. 1.5 m
2. 1.8 m
3. 1 m
4. 2 m

Q. Two wires of different materials, each $2m$ long and of diameter $2mm$, are joined in series to form a composite wire. What force will produce a total extension of $0.9mm$ ? ($Y_1=2\times {10}^{11}Pa$ and $Y_2=6\times {10}^{11}Pa$)

1. $282.6N$
2. $212N$
3. $319.8N$
4. $382.6N$